0,2

E_8 is also the Barnes-Wall lattice in 8 dimensions.

Expansion of Ramanujan's function Q(q)=12g2 (Weierstrass invariant).

D. Bump, Automorphic Forms..., Camb., 1997 p. 29.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.

S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.

S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.

Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978

Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994

N. J. A. Sloane, Table of n, a(n) for n = 0..1000

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares

G. Nebe and N. J. A. Sloane, Home page for E_8 lattice

H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms

S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Seven Staggering Sequences.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Eisenstein series

Index entries for sequences related to Barnes-Wall lattices

Can also be expressed as E4(q) = 1 + 240 sum_{i=1}^infinity i^3 q^i/(1-q^i) - Gene Ward Smith (genewardsmith(AT)gamil.com), Aug 22 2006

1 + 240*Sum ( sigma_3 (m) * q^2m ), m = 1..inf, where sigma_3 (m) is the sum of the cubes of the divisors of m (A001158).

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2+33*v^2+256*w^2-18*u*v+16*u*w-288*v*w . - Michael Somos Jan 05 2006

Expansion of (phi(-q)^8 -(2phi(-q)phi(q))^4 +16phi(q)^8) in powers of q where phi() is a Ramanujan theta function.

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)= +u1^2 +16*u2^2 +81*u3^2 +1296*u6^2 -14*u1*u2 -18*u1*u3 +30*u1*u6 +30*u2*u3 -288*u2*u6 -1134*u3*u6 . - Michael Somos Apr 15 2007

G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u, v, w)= +u^3*v +9*w*u^3 -84*u^2*v^2 +246*u*v^3 -253*v^4 -675*w*u^2*v +729*w^2*u^2 -4590*w*u*v^2 +19926*w*v^3 -54675*w^2*u*v +59049*w^3*u +531441*w^3*v -551124*w^2*v^2 . - Michael Somos Apr 15 2007

with(numtheory); E := proc(k) local n, t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..60); series(t1, q, 60); end; E(4);

(PARI) a(n)=if(n<1, n==0, 240*sigma(n, 3))

Cf. A046948, A000143, A108091 (eighth root).

Cf. A001158.

A008410 is convolution of this sequence with itself. A008411 is convolution of this sequence with A008410.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

Adjacent sequences: A004006 A004007 A004008 this_sequence A004010 A004011 A004012

Sequence in context: A092000 A124352 A049335 this_sequence A005950 A004536 A008340

nonn,easy,nice

njas